Probability means chances of occurrence of events. It is a way of measuring uncertainty.
For an event \(E\), the probability of getting a favourable outcome is given by:
The probability of event \(E\) is shown as \(P(E)\).
The probability of event \(E\) is shown as \(P(E)\).
\(P(E) =\) \(\frac{\text{The number of favourable outcomes}}{\text{The number of elements in the sample space}}\)
Complimentary events and probability
We denote the event that is 'not \(E\)' by . This is called the complement event of event \(E\).
So, \(P(E) + P(\text{not E}) = 1\).
Important!
i) The probability of each event lies between \(0\) and \(1\).
ii) The sum of all the probabilities is \(1\).
iii) \(E_1\), \(E_2\), . . .\(E_n\) covers all the possible outcomes of a trial.
ii) The sum of all the probabilities is \(1\).
iii) \(E_1\), \(E_2\), . . .\(E_n\) covers all the possible outcomes of a trial.
Algebra of events:
\(P(A \cup \overline A) = S\)
\(P(A \cap \overline A) = \phi\)
If the events \(A\) and \(B\) are mutually exclusive, then \(P(A \cup B) = P(A) + P(B)\).
\(P(\text{Union of events}) = \sum(\text{ Probability of events})\)
If \(A\) and \(B\) are two events associated with a random experiment then,
(i) \(P(A \cap \overline B)\) \(=\) \(P(\text{only A})\) \(=\) \(P(A) - P(A \cap B)\)
(ii) \(P(\overline A \cap B)\) \(=\) \(P(\text{only B})\) \(=\) \(P(B) - P(A \cap B)\)
(iii) If \(A\) and \(B\) are any two non mutually exclusive events then:
\(P(A \cup B)\) \(=\) \(P(A)\) \(+\) \(P(B)\) \(-\) \(P(A \cap B)\)
(iv) If \(A\), \(B\)and \(C\) are any three non mutually exclusive events then:
\(P(A \cup B \cup C)\) \(=\) \(P(A)\) \(+\) \(P(B)\) \(+\) \(P(C)\) \(-\) \(P(A \cap B)\) \(-\) \(P(B \cap C)\) \(-\) \(P(A \cap C)\) \(+\) \(P(A \cap B \cap C)\)